The DOI matches its Crossref record; 41 of the 41 references it cites were found in a registry or an index.
Introduction to bifurcation theory
Open accessAbstract
The theory of bifurcation from equilibria based on center-manifold reduction and Poincar\'e-Birkhoff normal forms is reviewed at an introductory level. Both differential equations and maps are discussed, and recent results explaining the symmetry of the normal form are derived. The emphasis is on the simplest generic bifurcations in one-parameter systems. Two applications are developed in detail: a Hopf bifurcation occurring in a model of three-wave mode coupling and steady-state bifurcations occurring in the real Landau-Ginzburg equation. The former provides an example of the importance of degenerate bifurcations in problems with more than one parameter and the latter illustrates new effects introduced into a bifurcation problem by a continuous symmetry.
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Cites
Applications of Lie Groups to Differential Equations
Olver, P. J. (1986) · Graduate Texts in Mathematics · doi:10.1007/978-1-4684-0274-2 · Cited by 3,773
Quantitative universality for a class of nonlinear transformations
Feigenbaum, M. J. (1978) · Journal of Statistical Physics · doi:10.1007/bf01020332 · Cited by 3,589
Differential Equations, Dynamical Systems, and Linear Algebra
(1974) · Pure and Applied Mathematics · doi:10.1016/s0079-8169(08)x6044-1 · Cited by 3,091
Singularities and Groups in Bifurcation Theory
Golubitsky, M., Stewart, I., & Schaeffer, D. G. (1988) · Applied Mathematical Sciences · doi:10.1007/978-1-4612-4574-2 · Cited by 2,958
Introduction to Applied Nonlinear Dynamical Systems and Chaos
Wiggins, S. (1990) · Texts in Applied Mathematics · doi:10.1007/978-1-4757-4067-7 · Cited by 2,784
Ruelle, D., & Takens, F. (1971) · Communications in Mathematical Physics · doi:10.1007/bf01646553 · Cited by 2,769
Hirsch, M. W. (1976) · Graduate Texts in Mathematics · doi:10.1007/978-1-4684-9449-5 · Cited by 2,520
VIII. Stability of a viscous liquid contained between two rotating cylinders
Taylor, G. I. (1923) · Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character · doi:10.1098/rsta.1923.0008 · Cited by 2,502
The Hopf Bifurcation and Its Applications
Marsden, J. E., & McCracken, M. (1976) · Applied Mathematical Sciences · doi:10.1007/978-1-4612-6374-6 · Cited by 2,300
Applications of Centre Manifold Theory
Carr, J. (1981) · Applied Mathematical Sciences · doi:10.1007/978-1-4612-5929-9 · Cited by 1,909
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Cross, M. C., & Hohenberg, P. C. (1993) · Reviews of Modern Physics · doi:10.1103/revmodphys.65.851 · Cited by 7,859
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Chen, L. Y., Goldenfeld, N., & Oono, Y. (1994) · Physical Review Letters · doi:10.1103/physrevlett.73.1311 · Cited by 260
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Metadata
- Authors
- Crawford, J. D.
- Venue
- Reviews of Modern Physics
- Publisher
- American Physical Society (APS)
- Published
- 1991-10-01
- Volume
- 63
- Issue
- 4
- Pages
- 991-1037
- ISSN
- 0034-6861 (ISSN-L)
- 1539-0756
- OpenAlex
- W2062528822
- Cited by
- 419
- References
- 41
- FWCI
- 15.88
- Type
- journal article
- Language
- en
- Open access
- green
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